A new shopping mall records $120$ total shoppers on their first day of business. Each day after that, the number of shoppers is $10\%$ more than the number of shoppers the day before. What is the total number of shoppers that visited the mall in the first $7$ days? Round your final answer to the nearest integer.
Notice that the daily counts of shoppers form a geometric sequence. The total number of shoppers after $ n$ days is the ${\text{sum}}$ of the first $n$ terms in the sequence. This is called a geometric series. This is the formula for that sum: $ S={a}\left(\dfrac{1-{r}^{ n}}{1-{r}}\right)$ where ${a}$ is the first term and ${r}$ is the common ratio. We can use this formula, along with the given information, to find the value of the sum, $ S$. Using the given information We are given that there were ${120}$ total shoppers on the first day. This is the first term $ a$. We are given that the number of shoppers each day is ${10\% \text{ more}}$ than the number of shoppers the day before. So we'll use a common ratio of ${1.10}$ for $ r$. There are ${7}$ days in the series. This is the number of terms $ n$. We want to find the total number of shoppers. This is the sum $ S$. Finding the sum $\begin{aligned} S&={120} \cdot \dfrac{1-\left({1.10}\right)^{{7}}}{1-\left({1.10}\right)} \\\\ &\approx{120} \cdot \dfrac{-0.9487}{-0.10} \\\\ & \approx 1138.46 \text{ shoppers} \end{aligned}$ Answer To the nearest integer, the total number of shoppers that visited the mall in the first $7$ days is $1138$ shoppers.